Coalgebraic Modal Logic: Fixpoints and Nested Modalities

Summary

Coalgebraic Logic is a paradigm that subsumes a large class of
structurally different modal logics, such as e.g. Coalition Logic,
graded and probabilistic modal logic and conditional logic,
classical and monotone modal logic as well as the modal logic K. The
goal of this project is to extend the scope of the coalgebraic
approach by considering two extensions to the basic setup: frame
conditions, in particular those that feature nested modalities, and
fixpoints.

Research Description

Modal logic is a formal framework for reasoning about change.
Applications of modal logics are abundant in computer
science and related disciplines, and a multitude of different
logics have been studied in a variety of application contexts.
Apart from classical applications in the field of concurrent,
mobile and probabilistic systems, modal logics are used in
artificial intelligence, e.g. in the context of reasoning with
uncertainty and -- in the shape of
description logics -- in the field of knowledge representation.
Modal logics are also employed to reason about games and
coalitional power in multi-agent systems. In economics, they
have been used to describe probabilistic information shared by
interacting agents whereas e.g. deontic logic, the logic of
obligation and permission is studied in philosophy.
The study of all of these logics centres around a number of
recurring questions, including completeness (``are all valid
statements formally derivable?''), decidability (``is the logic
amenable to automated reasoning?'') and complexity (``what
resources are required to mechanise the logic?''), which are
usually studied for each logic individually.
Rather than addressing these questions in a per-logic basis, it
is clearly desirable to conduct the studies in a uniform
framework that encompasses the greatest possible number of
concretely given logics as special cases.
The proposed research investigates and extends a generic
framework -- coalgebraic modal logic -- that encompasses a
large class of modal logics as specific examples, including all
instances mentioned above. The genericity of the approach
will lead to software tools that are not only more reliable but
also and easier to design, implement and to maintain.

Objectives

Development of a coalgebraic semantics for modal logics that
incorporate nested modalities and/or fixpoints